There's another strange assumption (to me at least) in the proof of Proposition
1.88 on page 24. In the process of proving the adjointness condition (in the
second-last paragraph), they use Eq. (1.6) from Proposition 1.81. But the whole
point of Proposition 1.81 is that Eq. (1.6) is just equivalent to the existence
of a Galois connection. Is it not a bit circular to assume the existence of a
Galois connection when attempting to prove the adjointness condition?